Currently proposed high speed transmission systems over optical fiber (e.g. 100 Gbps and beyond) use multiple bits per symbol as well as multiple polarizations in order to reduce cost and complexity of the design. Disadvantageously, high speed transmission over optical fiber suffers from a number of well known impairments including polarization-mode dispersion (PMD) where a signal on one polarization at a receiver is a mixture of the polarization signals transmitted, chromatic dispersion (CD) where the signal is subjected to a parabolic increasing phase distortion along the fiber, polarization gain imbalance where the gain of the two polarizations is not the same, and polarization delay imbalance where the travel time of the two polarizations is not the same. In a typical high speed transmission system implementation, two optical polarizations may be used with Quadrature Amplitude Modulation (QAM) on two orthogonal carriers on each polarization. Also, Quadrature Phase Shift Keying (QPSK) with four phases is a subset of such modulations. At a receiver of such a system, the two polarizations are typically recovered in an optical module where the quadrature signals are demodulated to baseband and converted to two quadrature electrical signals for each polarization. These four electrical signals are then transmitted to four analog to digital converters (ADC) followed by further processing in the digital domain.
Of note, the two demodulated signals of each polarization through the two ADCs are typically not fully orthogonal. While these impairments can be minimized using careful analog design, they cannot be completely eliminated, and their effect is a degradation of performance that increases quite fast as the magnitude of these impairments increases (see, e.g. I. Fatadin et al, “Compensation of Quadrature Imbalance in an Optical QPSK Coherent receiver”, IEEE Photonics Technology Letters, Vol. 20, No. 20, Oct. 15, 2008, pp 1733-1735). Conventional systems and methods for compensating the angle and magnitude imbalance introduced by the demodulator have been proposed, see, e.g., Fatadin et al.; C. S. Petru et al., “Impact of Transmitter and Receiver Imperfections on the Performance of Coherent Optical QPSK Communication Systems”, 21st Meeting of the IEEE Lasers and Electro Optics Society, November 2008, pp 410-411; A. Tarighat et al., “Compensation schemes and Performance Analysis of IQ Imbalance in OFDM receivers”, IEEE Trans on Signal Processing, Vol 53, No 8, August 2005, pp 3257-3267; and M. Valkama et al., “Advanced Methods for IQ Imbalance Compensation in Communication Systems”, IEEE Transactions on Signal Processing, Vol. 53, No. 10, October 201, pp 2335-2344. However, most of these methods deal with Orthogonal frequency-division multiplexing (OFDM) systems where multiple frequency tones are used to carry the information and the compensation is applied to these tones, typically in the frequency domain. Fatadin et al. deal with determining a compensation matrix directly from the correlations of the received data. Tarighat et al. propose a Least Mean Square (LMS) technique for updating the correction matrix based on transmitted training symbols either during a separate training period or as part of the transmission, leading to a loss of efficiency.
Additionally, the delay of the two demodulated signals of each polarization through the ADCs is not exactly equal. Similar to the orthogonal impairments, these impairments may be minimized using careful analog design but they cannot be completely eliminated, and their effect is a degradation of performance that increases quite fast as the magnitude of these impairments increases (see, e.g. T. Tanimura et al., “A Simple Digital Skew Compensator for Coherent Receiver”, ECOC 2009, 20-24 September, 2009, Paper 7.3.2). Conventional systems and methods for compensating for the time delay between quadrature paths using an finite impulse response (FIR) filter have been proposed in Tanimura et al. Tanimura et al. only deal with implementing a compensating interpolator for benefits obtained in high CD and PMD systems. However, Tanimura et al. do not give a method of deriving necessary delay parameters from the received data in the presence of such impairments.